It's sometimes nearly impossible to distinguish whether a piece of music is an Irish folk song or a pirate-era sea shanty. Why do these two styles of music have so much in common? Why such a strong Irish influence in sea shanties?
I wonder if you could clarify for me a bit of what your thinking is here with this question? I do think you are on to something, but the specific examples you gave make me think we do not have quite the same concepts in mind. For example. Rocky Road to Dublin is from the tail end of the 19th century, and Bully in the Alley wasn't collected until 1915, suggesting it's a later creation as well. So neither of these are really "pirate era" to begin with.
Then also, Bully in the Alley doesn't sound Irish to me at all and this is really where I want to clarify. Can you describe a bit of exactly what you think sounds Irish there, or give other examples of songs you think illustrate the question? I have a suspicion this is a music theory question in the end, but I'd like to know more before potentially going any further.